The QUANTSELECT Procedure

Quantile Process Regression

You can specify QUANTILE=PROCESS in the MODEL statement to perform quantile process regression. Quantile process regression fits quantile regression models for the entire range of quantile levels from 0 to 1. Because a quantile function is the inverse of its cumulative distribution function, quantile process regression can estimate the entire distribution of a response variable conditional on its covariates.

Because of the piecewise linearity of the check loss function, the optimal quantile regression solution ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis is a step function in tau element-of left-bracket 0 comma 1 right-bracket. In other words, given any optimal solution ModifyingAbove bold-italic beta With caret left-parenthesis tau Superscript asterisk Baseline right-parenthesis, there exists an optimal quantile-level range left-bracket tau 1 comma tau 2 right-bracket such that ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis equals ModifyingAbove bold-italic beta With caret left-parenthesis tau Superscript asterisk Baseline right-parenthesis is optimal for any tau element-of left-bracket tau 1 comma tau 2 right-bracket. This step-function property can simplify integration computation in quantile process regression. For example, to estimate conditional mean by using quantile process regression, you can substitute integration by using the summation

upper E left-parenthesis upper Y vertical-bar bold upper X equals bold x right-parenthesis equals integral Subscript 0 Superscript 1 Baseline bold x ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis d tau equals bold x sigma-summation Underscript i equals 1 Overscript s Endscripts left-parenthesis tau Subscript i plus 1 Baseline minus tau Subscript i Baseline right-parenthesis ModifyingAbove bold-italic beta With caret Subscript i

where tau 1 equals 0, tau Subscript s plus 1 Baseline equals 1, and ModifyingAbove bold-italic beta With caret Subscript i is the optimal solution for quantile range left-bracket tau Subscript i Baseline comma tau Subscript i plus 1 Baseline right-bracket.

If you specify the N=ALL suboption in the QUANTILE=PROCESS option, PROC QUANTSELECT outputs ModifyingAbove bold-italic beta With caret equals sigma-summation Underscript i equals 1 Overscript s Endscripts left-parenthesis tau Subscript i plus 1 Baseline minus tau Subscript i Baseline right-parenthesis ModifyingAbove bold-italic beta With caret Subscript i as the mean parameter estimates in the parameter estimates table. If you request the "Parameter Estimates for Quantile Process" table, PROC QUANTSELECT outputs parameter estimates in the following quantile-level grid:

StartSet 0 comma StartFraction tau 1 plus tau 2 Over 2 EndFraction comma StartFraction tau 2 plus tau 3 Over 2 EndFraction comma ellipsis comma StartFraction tau Subscript s Baseline plus tau Subscript s plus 1 Baseline Over 2 EndFraction comma 1 EndSet

For more information about the "Parameter Estimates for Quantile Process" table, see Parameter Estimates for Quantile Process. PROC QUANTSELECT also uses this grid to estimate observation quantile levels.

If you specify N=n, PROC QUANTSELECT approximates the quantile process regression in the following quantile-level grid:

StartSet 0 comma StartFraction 1 Over n plus 1 EndFraction comma StartFraction 2 Over n plus 1 EndFraction comma ellipsis comma 0.5 comma ellipsis comma StartFraction n Over n plus 1 EndFraction comma 1 EndSet

When N=n, PROC QUANTSELECT also approximates integrations by using the linear interpolation method, which defines

ModifyingAbove bold-italic beta With caret left-parenthesis tau right-parenthesis equals StartFraction tau minus tau 1 Over tau 2 minus tau 1 EndFraction ModifyingAbove bold-italic beta With caret left-parenthesis tau 2 right-parenthesis plus StartFraction tau 2 minus tau Over tau 2 minus tau 1 EndFraction ModifyingAbove bold-italic beta With caret left-parenthesis tau 1 right-parenthesis

Here,tau 1 and tau 2 denote two consecutive quantile levels in the quantile-level grid that satisfy tau element-of left-bracket tau 1 comma tau 2 right-bracket.

Last updated: December 09, 2022